Continuous Random Variables

According to the properties of continuous probability density functions, what is the probability that a continuous random variable X takes on any single individual value?

What is the total area under the curve of a probability density function and above the x-axis?

The amount of time a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive. What is the probability that a person waits fewer than 12.5 minutes?

For the uniform distribution X ~ U(0, 12), what is the theoretical standard deviation?

The length of a phone call, in minutes, is an exponential random variable with a mean of eight minutes. What is the decay parameter (m)?

What is the key characteristic of the memoryless property of the exponential distribution?

On average, a certain computer part lasts ten years, and its lifetime is exponentially distributed. What is the probability that this computer part lasts more than 7 years?

What is the relationship between the mean and standard deviation in an exponential distribution?

If the longevity of a light bulb is exponential with a mean lifetime of eight years, and a bulb has already lasted 12 years, what is the probability that it will last a total of over 19 years?

How does the Poisson distribution relate to the exponential distribution?

A continuous probability function is defined as f(x) = 1/12 for 0 <= x <= 12. What is the probability P(0 < x < 12)?

The total duration of baseball games in a typical season is uniformly distributed between 447 and 521 hours inclusive. What is the probability density function f(x) for this distribution?

The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with an average amount of time equal to eight minutes. What is the probability that a spouse will purchase a card in fewer than four minutes?

If a random variable is continuous, its outcomes are determined by what process?

What is the primary difference in the graphical representation of probability between a discrete distribution like the Poisson and a continuous distribution like the Uniform?

For a uniform distribution, if the lowest value 'a' is 10 and the highest value 'b' is 30, what is the height of the probability density function f(x)?

The age of cars in a staff parking lot is uniformly distributed from 0.5 years to 9.5 years. What is the average age of the cars in the lot?

At a 911 call center, calls come in at an average rate of one call every two minutes. Assuming an exponential distribution, what is the probability that after a call is received, it takes more than three minutes for the next call to occur?

For a continuous random variable X, the probability P(c < X < d) is equivalent to which of the following?

The time spent waiting between events is modeled using the exponential distribution, with an average of 30 customers per hour arriving at a store. On average, how many minutes elapse between two successive arrivals?

A distribution is given as X ~ U(0, 12). What is P(x > 9)?

For an exponential distribution, what does the decay parameter 'm' measure?

If a distribution is given as X ~ Exp(0.2), what is the mean of the distribution?

Suppose the amount of time a postal clerk spends with a customer is exponentially distributed with a mean of four minutes. What is the probability that a clerk spends between four and five minutes with a randomly selected customer?

A subway train on the Red Line arrives every eight minutes during rush hour, following a uniform distribution. What is the probability that a commuter waits between three and four minutes?

The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of 150 dollars. What is the probability that a car required over 300 dollars for maintenance during its first year?

The age of a first grader on September 1 is uniformly distributed from 5.8 to 6.8 years. What is the standard deviation of their ages?

What is the function that gives the area to the left of a value for a continuous random variable called?

A continuous probability function f(x) is restricted to the portion between x = 0 and 7. What is P(x = 10)?

Suppose the useful life of a car battery, in months, decays with parameter 0.025. On average, how long would you expect one car battery to last?

The number of days ahead travelers purchase their airline tickets can be modeled by an exponential distribution with the average amount of time equal to 15 days. What is the probability that a traveler will purchase a ticket fewer than ten days in advance?

A continuous probability function is f(x) = 1/3, restricted to 1 <= x <= 4. What does P(x > 8/3) describe?

For a uniform distribution X ~ U(a, b), what is the mean?

The number of miles driven by a truck driver falls between 300 and 700 and follows a uniform distribution. What is the probability that the truck driver goes between 400 and 650 miles in a day?

The average lifetime of a certain new smartphone is three years, and it follows an exponential distribution. What is the probability that the smartphone will fail within two years of the date of purchase?

In a uniform distribution, what does it mean for outcomes to be 'equally likely'?

A distribution is given as X ~ Exp(0.75). What is the standard deviation?

The value of a stock varies each day from 16 dollars to 25 dollars with a uniform distribution. Given that the stock is greater than 18 dollars, what is the probability that it is more than 21 dollars?

The median lifetime of a smartphone that follows an exponential distribution with a mean of three years is:

In the context of the exponential distribution, which statement best describes the relationship between the mean and the median?

A continuous random variable is defined by the probability density function f(x) = 1/5 for 0 <= x <= 5. What is P(x < 0)?

The age of cars in a staff parking lot is uniformly distributed from 0.5 to 9.5 years. What is the probability that a randomly chosen car is less than four years old?

If a distribution is given as X ~ Exp(0.75), what is the 30th percentile?

How is the cumulative distribution function (CDF) used to find the probability that a continuous random variable X falls between c and d, where c < d?

If seventy percent of car batteries last at least a certain number of months, and their lifetime follows an exponential distribution with decay parameter 0.025, what is that number of months?

Which of the following is NOT a property of the exponential distribution?

A random number generator picks a number from one to nine in a uniform manner. What is the probability that the number is greater than 5, given that it is greater than 3?

On average, a pair of running shoes can last 18 months if used every day, and the lifetime is exponentially distributed. Eighty percent of running shoes last at most how long if used every day?

The uniform distribution is often referred to as the rectangular distribution. Why?

The time, in minutes, until the next bus departs a major bus depot follows a distribution with f(x) = 1/20, where x goes from 25 to 45 minutes. What is the probability that the time is at most 30 minutes?